Ramification of stream networks
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Citation
Devauchelle, O., A. P. Petroff, H. F. Seybold, and D. H.
Rothman. “Ramification of stream networks.” Proceedings of the
National Academy of Sciences 109, no. 51 (December 18, 2012):
20832-20836.
As Published
http://dx.doi.org/10.1073/pnas.1215218109
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National Academy of Sciences (U.S.)
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Final published version
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Thu May 26 08:55:55 EDT 2016
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http://hdl.handle.net/1721.1/80312
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Detailed Terms
Ramification of stream networks
Olivier Devauchelle1, Alexander P. Petroff1, Hansjörg F. Seybold1, and Daniel H. Rothman1,2
Lorenz Center and Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139
Edited† by Nigel Goldenfeld, University of Illinois at Urbana–Champaign, Urbana, IL, and approved November 6, 2012 (received for review August 31, 2012)
The geometric complexity of stream networks has been a source of
fascination for centuries. However, a comprehensive understanding
of ramification—the mechanism of branching by which such networks grow—remains elusive. Here we show that streams incised
by groundwater seepage branch at a characteristic angle of 2π/5 =
72°. Our theory represents streams as a collection of paths growing
and bifurcating in a diffusing field. Our observations of nearly 5,000
bifurcated streams growing in a 100 km2 groundwater field on the
Florida Panhandle yield a mean bifurcation angle of 71.9° ± 0.8°. This
good accord between theory and observation suggests that the network geometry is determined by the external flow field but not, as
classical theories imply, by the flow within the streams themselves.
river networks
| network growth | Laplacian growth
lows shape landscapes and landscapes shape flows (1). In time,
streams form, channelize, and organize into highly ramified
networks. These branched structures, which have attracted interest
at least since the time of Leonardo da Vinci (2), exhibit a striking
geometry with similar shapes repeating on all scales (3, 4). However, precisely how stream networks grow and create these geometries remains elusive (3, 5, 6).
To obtain a better understanding of network growth, here we
investigate the ramification of channels fed by springs. The reemergence of groundwater at springs is often sufficient to incise
channels (1, 7–9). It can also contribute to the initiation and advance of the channel heads in conjunction with erosion due to
overland flow (10, 11). Channelization driven by such “seepage
erosion” is therefore of considerable practical interest in hydrology. It has also been extensively considered as a mechanism for the
erosive incision of channelized features on Mars (12–17).
Our study of seepage erosion is, however, motivated by the
fundamental problem of network branching (18). Unlike overland
flow, groundwater flow is only weakly sensitive to overlying topography. The flow of water to the channel head is straightforwardly
characterized as a problem of fluid mechanics: subsurface flow is
governed by Darcy’s law (19), which in Dupuit’s approximation (20)
leads to simple linear approximations that capture the natural
problem rather well (21, 22). Because erosion is typically slow
compared with relaxation of the water table, topography enters the
calculation of groundwater flux only via the specification of
boundary conditions at channels (when the overlying permeability is
uniform). The characterization of flow into springs is therefore well
described mathematically. However, behind this veneer of simplicity lies the vast complexity of ramified drainage networks (23).
One thus expects that insights made possible by specialization to the
seepage problem should prove useful for analyzing problems of
channelization due principally to overland flow (3, 10, 24–28).
F
Physical Picture
We focus on the valley network of Fig. 1, located on the Florida
Panhandle. The streams at the base of these valleys have been cut
through unconsolidated sand (23, 29). Each valley head contains
a spring through which groundwater seeps to the surface. The
resulting surficial flow, which derives entirely from groundwater
seepage (23, 29, 30), then carries sand to the network’s outlet. Over
geologic time, valley heads advance and the network ramifies (23).
We seek an understanding of ramification by studying the angle at
which valley heads bifurcate into distinct branches.
20832–20836 | PNAS | December 18, 2012 | vol. 109 | no. 51
Zooming in more closely, Fig. 2 shows two specific bifurcated
tips. One sees that the resulting streams are not necessarily
straight. They can meander or curve due to heterogeneities arising,
for example, from vegetation and interactions with other streams.
We nevertheless characterize a nascent bifurcation as straight
segments of a Y-shaped path. In the Florida network, the slopes of
streams are small (∼1%), and as clearly seen in Fig. 2, their widths
are negligible compared with the scale of the network. We therefore assume that the individual segments of the Y-shaped path are
infinitesimally thin and lie in the horizontal plane.
When a single tip splits into two springs, the groundwater field
around the newly formed springs is influenced chiefly by competition
between the springs themselves, rather than by the distant network,
suggesting that there should be much to learn from the groundwater
flow field surrounding a single bifurcated tip. Fig. 3 illustrates the
flow field around the Y-shaped bifurcation, at different opening
angles α. The paths (streamlines) along which groundwater flows
into the tips are marked in red. A mathematical expression for these
paths is given below. At this juncture, their pictorial expression in
Fig. 3 suffices to illustrate basic physical mechanisms. We see that
when α is small, the nascent tips compete with one another for
groundwater, causing them to grow outward. When α is large, the
nascent tips compete with the parent stream, causing them to grow
inward. At a special angle α*, the tips grow forward without turning.
Because the lack of curvature suggests an independence of scale, we
expect that real networks should bifurcate at the angle α*.
Theory
We seek the exact value of α*. Our theory is built upon the
following hypothesis: streams grow forward in the direction from
which groundwater enters their tips (13).
To find the groundwater field around the bifurcated tip, we
note that the height h of the water table above an impermeable
layer is a solution of the Poisson equation (21, 22, 30)
∇2 h2 ðx; yÞ = −
2P
;
K
[1]
where P is the mean precipitation rate and K the hydraulic conductivity. Groundwater flows into the stream network along the
paths of steepest descent down h2 and intersects the stream at the
elevation of the channel profile. The network ramifies when new
springs form as the result of a bifurcation of an existing spring.
We make two additional observations. First, rain falling immediately around a bifurcated spring represents a negligible fraction
of the water draining into it. Specifically, the influence of precipitation P in Eq. 1 can be neglected whenever the length of the
nascent branch is small relative to the typical linear dimension of
a drainage area. The height of the water table near the stream is
then a solution of Laplace’s equation, expressed here as
Author contributions: O.D., A.P.P., H.F.S., and D.H.R. designed research, performed research, analyzed data, and wrote the paper.
The authors declare no conflict of interest.
†
This Direct Submission article had a prearranged editor.
Freely available online through the PNAS open access option.
1
O.D., A.P.P., H.F.S., and D.H.R. contributed equally to this work.
2
To whom correspondence should be addressed. E-mail: dhr@mit.edu.
www.pnas.org/cgi/doi/10.1073/pnas.1215218109
∇2 h2 = 0:
[2]
Our second observation concerns boundary conditions: when the
lengths of incipient streams are sufficiently small, the influence of
the remainder of the network can be neglected and the branches
feel an effectively infinite Laplacian field. The growth of the Florida
network is therefore a natural manifestation of the growth of onedimensional paths in a two-dimensional harmonic field (31, 32).
We proceed to solve Eq. 2 for the shape of the water table
around a Y-shaped bifurcation that completely absorbs any
groundwater that reaches it. The problem is straightforwardly
analyzed in the complex plane (31, 33–36), where the coordinates
of the point (x,y) are represented by the complex number z = x + iy,
A
where i2 = − 1. We describe the shape of a bifurcated head as
follows. First, we choose a coordinate system in which the main
channel lies on the negative imaginary axis z = iy, where y < 0. The
nascent branches then lie on the lines
σ ± ðrÞ = re iðπ∓αÞ=2 ;
0 < α < 2π;
[3]
where r > 0. These streams meet at the origin z = 0. Notably, the
only length scale needed to describe this bifurcation is the length
of the nascent streams, which we choose to be unity. The growth
process of an incipient bifurcation has no characteristic length
scale, and therefore, we assume that during the first step of
growth the bifurcation maintains its initial shape. Consequently,
B
Fig. 2. (A) Slope map of valley heads in the Florida network. Water flows from top to bottom. Note the small incipient bifurcation in the uppermost valley
head. (B) A confluence of two streams at a channel bifurcation. Water flows out of the picture at the upper right. Note person for scale.
Devauchelle et al.
PNAS | December 18, 2012 | vol. 109 | no. 51 | 20833
EARTH, ATMOSPHERIC,
AND PLANETARY SCIENCES
Fig. 1. High-resolution topographic map of valley networks incised by groundwater flow, located near Bristol, Florida, on the Florida Panhandle. Streams at
the base of the valleys flow toward the Apalachicola River, located near the left boundary.
A
C
E
B
D
F
Fig. 3. Streamlines (black) of a groundwater field arriving from infinity to a Y-shaped bifurcation (blue), for different values of the opening angle α. The
streamlines entering tips are indicated in red. The shaded regions in the top row are magnified and shown in the bottom row. (A and B) When the bifurcation
angle is large, streams bend toward each other as they grow in the direction of the red streamline. (C and D) At a special angle α*, streams grow straight
without curving. (E and F) When the bifurcation angle is small, the streams grow outwards.
the length of the streams can be rescaled in time to reveal an
unchanging configuration, thus reducing the dynamical growth
problem to a problem of geometry.
Given this parameterization of the bifurcated tip, we solve for
the shape of the water table by conformal mapping (37). Using the
Schwarz–Christoffel transformation (37), one can find the complex function
gðwÞ = eiðπ−αÞ=2 wα=π
1−α=2π
2π − αw2
;
2π − α
[4]
which maps the upper half-plane w = u + iv to the region around the
bifurcated tips described in Eq. 3. This mapping, illustrated in Fig. 4,
relates the solution h2 ðzÞ of Eq. 2 around the bifurcated tip to the
~2
trivial solution h ðwÞ = Reð−iwÞ in the upper half of the mathematical plane, where the absorbing boundary is along the real axis.
In particular, the shape of the water table around the bifurcated tip is
h ðzÞ = Re −ig−1 ðzÞ ;
2
where g−1 is the inverse function of the mapping g given by Eq. 4.
Following our hypothesis that springs grow in the direction
from which groundwater flows into them, we must find the
streamlines flowing into the tips to determine the direction of
growth. As illustrated in Fig. 4, the streamlines flowing into the
springs in the mathematical (w) plane at w = ± 1 are mapped
from the vertical lines wðvÞ = ± 1 + iv. Consequently, in the
physical (z) plane, water flows into the spring at w = + 1 along
the curve
[5]
zðvÞ = gð1 + ivÞ = RðvÞeiðπ−αÞ=2 ;
[6]
where
RðvÞ = ð1 + ivÞ
α=π
2π − αð1 + ivÞ2
2π − α
!1−α=2π
:
[7]
The remaining streamline may be found by symmetry.
Eqs. 6 and 7 provide the shape of the streamline flowing into a
spring for an arbitrary value of α. We identify the special angle α*
Fig. 4. Mapping of the upper half of the mathematical plane w (Left) to the physical plane z (Right). The solid blue line is mapped from the real axis in the w plane
to the Y-shaped bifurcation in the z plane. The vertical red lines w ± ðvÞ in the w plane are mapped onto the streamlines g½w ± ðvÞ along which groundwater flows
into the springs of the bifurcated stream in the physical plane. The dashed light blue line is mapped from the w plane to a contour of constant h2 in the z plane.
20834 | www.pnas.org/cgi/doi/10.1073/pnas.1215218109
Devauchelle et al.
with the angle for which the streamline enters a tip without curvature. The curvature of RðvÞ is given by κ α ðvÞ = Im½R″=ðR′jR′jÞ.
After expanding around v = 0 up to first order, one obtains
5α − 2π 1
+ OðvÞ:
[8]
κ α ðvÞ =
8α
v
Discussion and Conclusion
The coupling of our growth hypothesis—streams grow in the direction from which flow enters tips (13)—with our simple model of
We find that the curvature vanishes at the origin only when α
equals
α* =
2π
= 728:
5
[9]
In all other cases, the curvature diverges like 1/v. A similar result
has been obtained by Hastings for the dielectric breakdown model
at the upper critical point (33). Related calculations may also be
found in refs. 34–36.
Observation
We now return our attention to the real world. We analyze the
valley networks of Fig. 1 and others immediately adjacent to it
using a high-resolution LIDAR (Light Detection And Ranging)
(38) map that extends over ∼100 km2 with a horizontal resolution
Devauchelle et al.
Fig. 6. Histogram of 4,966 measured junctions (blue points), compared with
a Gaussian distribution (red) with the sample mean and variance. The estimated mean bifurcation angle, hαi = 71:98 ± 0:88, is consistent with the prediction α = 2π=5 = 728.
PNAS | December 18, 2012 | vol. 109 | no. 51 | 20835
EARTH, ATMOSPHERIC,
AND PLANETARY SCIENCES
Fig. 5. Channel network extracted from the topography. Fig. 1 corresponds
roughly to the lower half of this network. Streams are colored according to their
Horton–Strahler order. The Apalachicola River is shown as a dark blue line. The
WGS 84 coordinates of the upper left corner are 308 26′ 59″N, 858 00′ 25″W.
of 1.2 m and a vertical resolution of about 5 cm. The combination
of extensive spatial coverage and high resolution allows us to
measure the angles of 4,966 channel junctions.
We first locate the streams. Streams stand out in a landscape
as sharply incised cuts through the otherwise smooth topography.
We thus find all grid points where the elevation contours are
sharply curved (39–41). Rills in the ravine walls are then eliminated by requiring the slope to be moderate at the channel
bottom. Finally, the resulting network is reduced to a collection
of one-pixel-wide paths. The streams are then interpolated from
the gridded data and indexed by their Horton–Strahler order
(10). Fig. 5 shows the resulting network.
We determine the direction of valley growth by approximating
each stream of a given Horton–Strahler order by a straight line
using the reduced major axis (42, 43). We then define α as the angle
between the two upstream segments of intersecting stream directions. Fig. 6 shows a histogram of the measured angles at all 4,966
junctions of our network. The mean angle hαi = 71:98 ± 0:88 (95%
C.I.) is unambiguously consistent with the prediction of Eq. 9.
Although the error in the estimate of the mean angle is reasonably small, the standard deviation of the histogram, 27.7°, is considerable. Because the typical standard error associated with each
angle measurement is only a few degrees, it would appear that
deflections of streams due to localized material heterogeneities,
landslides, vegetation, tree falls, etc., do not significantly contribute
to the large sample variance. We are instead led to identify aspects
of the bifurcation dynamics not addressed by our theory. First, we
note that streams need not initially ramify at 72°; more generally, we
expect that 72° represents a stable fixed point for growth in an
infinite Laplacian field (34, 44). However, groundwater is a finite
Poissonian field that reflects interactions with the rest of the
channel network. We therefore hypothesize that the wide variance
of the angle distribution reflects a combination of the transient
approach to 72° at early stages of bifurcation and a departure at late
stages as tips advance toward groundwater divides. In other words,
the self-similarity of the 72° bifurcation occurs in the limit of intermediate asymptotics (45, 46), whereas our measurements may
not. Curiously, diffusion-limited aggregation (47) exhibits a similar
large variance, but its angle histograms are asymmetric (48), unlike
the histogram of Fig. 6.
groundwater flow (21, 22) appears to capture all of the mechanisms required to explain the average angle at which streams split
in the Florida network. The ramification of the network is therefore entirely associated with the extrinsic groundwater field,
without regard to the flow of water and sediment within the
streams. This need not have been the case. For example, a widely
held view of stream network formation suggests that the fractal
geometry of stream networks is a consequence of the minimization
of energy dissipation within streams (3). In analogy with Murray’s
law for the growth of vascular networks (49), one could reason that
the bifurcation angle itself is a consequence of such a minimization
principle (50). If so, the angle would be a function of slope s and
discharge Q (50). In the Florida streams, theory and observation
provide the relation Q ∝ s−2 (51). Minimizing dissipation within
streams would then predict a bifurcation angle of 90° for two
streams of equal discharge and slope (50), in disagreement with our
ACKNOWLEDGMENTS. We thank Y. Couder, L. Kadanoff, M. MineevWeinstein, P. Szymczak, and G. Vasconcelas for helpful discussions and
K. Kebart and the Northwest Florida Water Management District for providing
high-resolution topographic maps. This work was supported by Department of
Energy Grant FG02-99ER15004. H.J.S. was additionally supported by the Swiss
National Science Foundation.
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20836 | www.pnas.org/cgi/doi/10.1073/pnas.1215218109
measurements. In contrast, we find that the mechanism generating
the Florida network’s geometric complexity is clearly independent
of flow within streams. Moreover, our theory requires no assumption of optimality.
Distinctions between intrinsic and extrinsic control exist not
only in studies of stream networks but also in investigations of
neuron dendrites (52), fungal hyphae (53), and other branching
problems. Because our results show how external and internal
controls can lead to quantitatively distinct geometric features,
we expect that similar analyses should be of use wherever the
provenance of ramification remains controversial.
Devauchelle et al.